Electron-positron pairs production in a macroscopic charged core

Classical and semi-classical energy states of relativistic electrons bounded by a massive and charged core with the charge-mass-radio Q/M and macroscopic radius R_c are discussed. We show that the energies of semi-classical (bound) states can be much smaller than the negative electron mass-energy (-mc^2), and energy-level crossing to negative energy continuum occurs. Electron-positron pair production takes place by quantum tunneling, if these bound states are not occupied. Electrons fill into these bound states and positrons go to infinity. We explicitly calculate the rate of pair-production, and compare it with the rates of electron-positron production by the Sauter-Euler-Heisenberg-Schwinger in a constant electric field. In addition, the pair-production rate for the electro-gravitational balance ratio Q/M = 10^{-19} is much larger than the pair-production rate due to the Hawking processes.

The energetics of this phenomenon can be understood as follow. The energy-level of the bound state 1S 1/2 can be estimated as follow, wherer is the average radius of the 1S 1/2 state's orbit, and the binding energy of this state Ze 2 /r > 2mc 2 . If this bound state is unoccupied, the bare nucleus gains a binding energy Ze 2 /r larger than 2mc 2 , and becomes unstable against the production of an electron-positron pair. Assuming this pair-production occur around the radiusr, we have energies of electron (ǫ − ) and positron (ǫ + ): where p ± are electron and positron momenta, and p − = −p + . The total energy required for a pair production is, which is independent of the potential V (r). The potential energies ±eV (r) of electron and positron cancel each other and do not contribute to the total energy (8) required for pair production. This energy (8) is acquired from the binding energy (Ze 2 /r > 2mc 2 ) by the electron filling into the bound state 1S 1/2 . A part of the binding energy becomes the kinetic energy of positron that goes out. This is analogous to the familiar case that a proton (Z = 1) catches an electron into the ground state 1S 1/2 , and a photon is emitted with the energy not less than 13.6 eV.
In this article, we study classical and semi-classical states of electrons, electron-positron pair production in an electric potential of macroscopic cores with charge Q = Z|e|, mass M and macroscopic radius R c .

II. CLASSICAL DESCRIPTION OF ELECTRONS IN POTENTIAL OF CORES
A. Effective potentials for particle's radial motion Setting the origin of spherical coordinates (r, θ, φ) at the center of such cores, we write the vectorial potential A µ = (A, A 0 ), where A = 0 and A 0 is the Coulomb potential. The motion of a relativistic electron with mass m and charge e is described by its radial momentum p r , total angular momenta p φ and the Hamiltonian, where the potential energy V (r) = eA 0 , and ± corresponds for positive and negative energies. The states corresponding to negative energy solutions are fully occupied. The total angular momentum p φ is conserved, for the potential V (r) is spherically symmetric. For a given angular momentum where v ⊥ is the transverse velocity, the effective potential energy for electron's radial motion is where ± indicates positive and negative effective energies, outside the core (r ≥ R c ), the Coulomb potential energy V (r) is given by Inside the core (r ≤ R c ), the Coulomb potential energy is given by where we postulate the charged core has a uniform charge distribution with constant charge density ρ = Ze/V c , and the core volume V c = 4πR 3 c /3. Coulomb potential energies outside the core (11) and inside the core (12) are continuous at r = R c . The electric field on the surface of the core, where the electron Compton wavelength λ e =h/(mc), the critical electric field E c = m 2 c 3 /(eh) and the parameter β is the electric potential-energy on the surface of the core in unit of the electron mass-energy.
B. Stable classical orbits (states) outside the core.
Given different values of total angular momenta p φ , the stable circulating orbits R L (states) are determined by the minimum of the effective potential E + (r) (10) (see Fig. 1), at which dE + (r)/dr = 0. We obtain stable orbits locate at the radii R L outside the core, for different p φ -values. Substituting Eq. (14) into Eq. (10), we find the energy of electron at each stable orbit, For the condition R L > ∼ R c , we have where the semi-equality holds for the last stable orbits outside the core R L → R c + 0 + . In the point-like case R c → 0, the last stable orbits are Eq. (15) shows that there are only positive or null energy solutions (states) in the case of a pointlike charge, which corresponds to the energy-spectra Eqs. (3,4,5) in quantum mechanic scenario.
While for p φ ≫ 1, radii of stable orbits R L ≫ 1 and energies E → mc 2 + 0 − , classical electrons in these orbits are critically bound for their banding energy goes to zero. We conclude that the energies (15) of stable orbits outside the core must be smaller than mc 2 , but larger than zero, E > 0. Therefore, no energy-level crossing with the negative energy spectrum occurs.
C. Stable classical orbits inside the core.
We turn to the stable orbits of electrons inside the core. Analogously, using Eqs. (10,12) and dE + (r)/dr = 0, we obtain the stable orbit radius R L ≤ 1 in the unit of R c , obeying the following equation, and corresponding to the minimal energy (binding energy) of these states There are 8 solutions to this polynomial equation (18), only one is physical, the solution R L that has to be real, positive and smaller than one. As example, the numerical solution to Eq. (18) is R L = 0.793701 for β = 4.4 · 10 16 and κ = 2.2 · 10 16 . In following, we respectively adopt non-relativistic and ultra-relativistic approximations to obtain analytical solutions.
First considering the non-relativistic case for those stable orbit states whose kinetic energy term characterized by angular momentum term p φ , see Eq. (10), is much smaller than the rest mass term mc 2 , we obtain the following approximate equation, and the solutions for stable orbit radii are, and energies, The consistent conditions for this solution are β 1/2 > κ for R L < 1, and β ≪ 1 for non-relativistic . As a result, the binding energies (22) of these states are mc 2 > E > 0, are never less than zero. These in fact correspond to the stable states which have large radii closing to the radius R c of cores and v ⊥ ≪ c.
Second considering the ultra-relativistic case for those stable orbit states whose the kinetic energy term characterized by angular momentum term p φ , see Eq. (10), is much larger than the rest mass term mc 2 , we obtain the following approximate equation, and the solutions for stable orbit radii are, which gives R L ≃ 0.7937007 for the same values of parameters β and κ in above. The consistent condition for this solution is β > κ ≫ 1 for R L < 1. The energy levels of these ultra-relativistic states are, and mc 2 > E > −1.5βmc 2 . The particular solutions E = 0 and E ≃ −mc 2 are respectively given by These in fact correspond to the stable states which have small radii closing to the center of cores To have the energy-level crossing to the negative energy continuum, we are interested in the values β > κ ≫ 1 for which the energy-levels (25) of stable orbit states are equal to or less than −mc 2 , As example, with β = 10 and κ = 2, R L ≃ 0.585, E min ≃ −9.87mc 2 . The lowest energy-level of electron state is p φ /(Ze 2 ) = κ/β → 0 with the binding energy, locating at R L ≃ (p φ c/Ze 2 ) 1/3 → 0, the bottom of the potential energy V in (0) (12).

A. Bohr-Sommerfeld quantization
In order to have further understanding, we consider the semi-classical scenario. Introducing the Planck constanth = h/(2π), we adopt the semi-classical Bohr-Sommerfeld quantization rule which are discrete values selected from continuous total angular momentum p φ in the classical scenario. The variation of total angular momentum ∆p φ = ±h in th unit of the Planck constant h, we make substitution in classical solutions that we obtained in section (II).
1. The radii and energies of stable states outside the core (14) and (15) become: where the electron Compton length λ =h/(mc).
2. The radii and energies of non-relativistic stable states inside the core (21) and (22) become: 3. The radii and energies of ultra-relativistic stable states inside the core (24) and (25) become: Note that radii R L in the second and third cases are in unit of R c .

B. Stability of semi-classical states
When these semi-classical states are not occupied as required by the Pauli Principle, the transition from one state to another with different discrete values of total angular momentum l (l 1 , l 2 and ∆l = l 2 − l 1 = ±1) undergoes by emission or absorption of a spin-1 (h) photon. Following the energy and angular-momentum conservations, photon emitted or absorbed in the transition have angular momenta p γ = p φ (l 2 ) − p φ (l 1 ) =h(l 2 − l 1 ) = ±h and energy E γ = E(l 2 ) − E(l 1 ). In this transition of stable states, the variation of radius is ∆R L = R L (l 2 ) − R L (l 1 ).
This state l =l > 0 is not protected by the Heisenberg indeterminacy principle from quantummechanically decaying inh-steps to the states with lower angular momenta and energies (correspondingly smaller radius R L (31)) via photon emissions. This clearly shows that the "Z = 137catastrophe" corresponds to R L → 0, falling to the center of the Coulomb potential and all semiclassical states (l) are unstable.
Then we consider the stability of semi-classical states against such transition in the case of charged cores R c = 0. Substituting p φ in Eq. (29) into Eq. (16), we obtain the selected semiclassical statel corresponding to the last classical stable orbit outside the core, Analogously to Eq. (37), the same argument concludes the instability of this semi-classical state, which must quantum-mechanically decay to states with angular momentum l <l inside the core, provided these semi-classical states are not occupied. This conclusion is independent of Zα-value.
We go on to examine the stability of semi-classical states inside the core. In the non-relativistic case (1 ≫ β > κ 2 ), the last classical stable orbits locate at R L → 0 and p φ → 0 given by Eqs. (21,22), corresponding to the lowest semi-classical state (33,34) with l = 0 and energy mc 2 > E > 0. In the ultra-relativistic case (β > κ ≫ 1), the last classical stable orbits locate at R L → 0 and p φ → 0 given by Eqs. (24,25), corresponding to the lowest semi-classical state (35,36) with l = 0 and minimal energy, This concludes that the l = 0 semi-classical state inside the core is an absolute ground state in both non-and ultra-relativistic cases. The Pauli principle assures that all semi-classical states l > 0 are stable, provided all these states accommodate electrons. The electrons can be either present inside the core or produced from the vacuum polarization, later will be discussed in details.
We are particular interested in the ultra-relativistic case β > κ ≫ 1, i.e., Zα ≫ 1, the energylevels of semi-classical states can be more profound than −mc 2 (E < −mc 2 ), energy-level crossings and pair-productions occur if these states are unoccupied, as discussed in introductory section.

IV. PRODUCTION OF ELECTRON-POSITRON PAIR
When the energy-levels of semi-classical (bound) states E ≤ −mc 2 (27), energy-level crossings between these energy-levels (25) and negative energy continuum (10) for p r = 0, as shown in Fig. 2.
The energy-level-crossing indicates that E (25) and E − (10) are equal, where angular momenta p φ in E (36) and E − (10) are the same for angular-momentum conservation.
The production of electron-positron pairs must takes place, provided these semi-classical (bound) states are unoccupied. The phenomenon of pair production can be understood as a quantummechanical tunneling process of relativistic electrons. The energy-levels E of semi-classical (bound) states are given by Eq. (36) or (27). The probability amplitude for this process can be calculated by a semi-classical WKB method [19]: where |p ⊥ | = p φ /r is transverse momenta and the radial momentum, The energy potential V (r) is either given by V out (r) (11) for r > R c , or V in (r) (12) for r < R c .
The To obtain a maximal WKB-probability amplitude (41) of pair production, we only consider the case that the charge core is bare and • the lowest energy-levels of semi-classical (bound) states: p φ /(Ze 2 ) = κ/β → 0, the location of classical orbit (24) R L = R b → 0 and energy (25) E → E min = −3βmc 2 /2 (28); • another classical turning point R n ≤ R c , since the probability is exponentially suppressed by a large tunneling length ∆ = R n − R b .
In this case (R n ≤ R c ), Eq. (42) becomes and p r = 0 leads to Using Eqs. (41,43,44), we have Dividing this probability amplitude by the tunneling length ∆ ≃ R n and time interval ∆t ≃ 2πh/(2mc 2 ) in which the quantum tunneling occurs, and integrating over two spin states and the transverse phase-space 2 dr ⊥ dp ⊥ /(2πh) 2 , we approximately obtain the rate of pair-production per the unit of time and volume, where E s = Ze/R 2 c is the electric field on the surface of the core and the Compton time τ =h/mc 2 . To have the size of this pair-production rate, we consider a macroscopic core of mass M = M ⊙ and radius R c = 10km, and the electric field on the core surface E s (13) is about the critical field (E s ≃ E c ). In this case, Z = α −1 (R c /λ) 2 ≃ 9.2 · 10 34 , β = Zαλ/R c = R c /λ ≃ 2.59 · 10 16 , and the rate (47) becomes which is exponentially small for R c ≫ λ. In this case, the charge-mass radio Q/(G 1/2 M ) = 2 · 10 −6 |e|/(G 1/2 m p ) = 8.46 · 10 −5 , where G is the Newton constant and proton's charge-mass radio It is interesting to compare this rate of electron-positron pair-production with the rate given by the Hawking effect. We take R c = 2GM/c 2 and the charge-mass radio Q/(G 1/2 M ) ≃ 10 −19 for a naive balance between gravitational and electric forces. In this case β = 1 2 (Q/G 1/2 M )(|e|/G 1/2 m) ≈ 10 2 , the rate (47) becomes, where the notation mM = R c /(2λ). This is much larger than the rate of electron-positron emission by the Hawking effect [23], since the exponential factor exp {−0.492(mM )} is much larger than exp {−8π(mM )}, where 2mM = R c /λ ≫ 1.

V. SUMMARY AND REMARKS
In this letter, analogously to the study in atomic physics with large atomic number Z, we study the classical and semi-classical (bound) states of electrons in the electric potential of a massive and charged core, which has a uniform charge distribution and macroscopic radius. We have found negative energy states of electrons inside the core, whose energies can be smaller than −mc 2 , and the appearance of energy-level crossing to the negative energy spectrum. As a result, quantum tunneling takes place, leading to electron-positron pairs production, electrons then occupy these semi-classical (bound) states and positrons are repelled to infinity. Assuming that massive charged cores are bare and non of these semi-classical (bound) states are occupied, we analytically obtain the maximal rate of electron-positron pair production in terms of the core radius, charge and mass. We find that this rate is much larger than the rate of electron-positron pair-production by the Hawking effect, even for very small charge-mass radio of the core given by the naive balance between gravitational and electric forces. orbits where E + has a minimum (15). All stable orbits are described by cp φ > Ze 2 . The last stable orbits are given by cp φ → Ze 2 + 0 + , whose radial location R L → 0 and energy E → 0 + . There is no any stable orbit with energy E < 0 and the energy-level crossing with the negative energy spectrum E − is impossible. All stable orbits inside the core are described by β > κ > 1. The last stable orbit is given by κ/β → 0, whose radial location R L → 0 and energy E → E min (28). We indicate that the energy-level crossing between bound state (stable orbit) energy at R L = R b and negative energy spectrum E − (25) at the turning point